Killing Horizons: Negative Temperatures and Entropy Super-Additivity

Many discussions in the literature of spacetimes with more than one Killing horizon note that some horizons have positive and some have negative surface gravities, but assign to all a positive temperature. However, the first law of thermodynamics then takes a non-standard form. We show that if one regards the Christodoulou and Ruffini formula for the total energy or enthalpy as defining the Gibbs surface, then the rules of Gibbsian thermodynamics imply that negative temperatures arise inevitably on inner horizons, as does the conventional form of the first law. We provide many new examples of this phenomenon, including black holes in STU supergravity. We also give a discussion of left and right temperatures and entropies, and show that both the left and right temperatures are non-negative. The left-hand sector contributes exactly half the total energy of the system, and the right-hand sector contributes the other half. Both the sectors satisfy conventional first laws and Smarr formulae. For spacetimes with a positive cosmological constant, the cosmological horizon is naturally assigned a negative Gibbsian temperature. We also explore entropy-product formulae and a novel entropy-inversion formula, and we use them to test whether the entropy is a super-additive function of the extensive variables. We find that super-additivity is typically satisfied, but we find a counterexample for dyonic Kaluza-Klein black holes.


Introduction
Since the early days of black hole thermodynamics there have been suggestions that the thermodynamic of the inner, Cauchy, horizons of charged and or rotating black holes should be taken more seriously than it has been [1][2][3][4][5][6][7][8][9][10]. With the development of String Theory approaches these suggestions have become more insistent [11][12][13][14][15][16][17]. This interest increased considerably with the observation that the product of the areas and hence entropies of the inner and outer horizon takes in many examples a universal form which should be quantised at the quantum level [18][19][20][21][22]. Some of these papers, and others, for example [22][23][24][25][26][27], encountered the same feature first noticed in [1]: the fact that with a conventional first law of thermodynamics the temperature of the inner horizon would be negative. The authors of [22] chose to resolve this issue by defining the temperature of the inner horizon to be the absolute value of the "thermodynamic" temperature, and proposing an appropriatelymodified first law on the inner horizon to compensate for this. In this paper we shall explore the consequences of adhering to the standard first law of thermodynamics for inner horizons, with the inevitable result that the temperature will be negative there.
In the derivation of the first law of black hole dynamics one finds, integrating in the region between the inner and outer horizons, that where κ ± are the surface gravities and A ± the areas of the outer and inner horizons respectively. (The contributions from the angular momentum and charge(s) are represented by the ellipses in this equation.) If, as turns out to be the case in the examples we consider, the signs of dA + and dA − are opposite for a given change in the black-hole parameters, then the signs of the surface gravities at κ + and κ − must be opposite too. The surface gravity is defined by evaluating on the horizon, where ℓ µ is the future-directed null generator of the horizon, which coincides with a Killing vector K µ on the horizon. One then finds that whilst κ is positive on the outer horizon, it is negative on the inner horizon. 1 Hawking showed that for an isolated 1 For example, in a static metric ds 2 = −h(r) dt 2 +dr 2 /h(r)+r 2 (dθ 2 +sin 2 θ dφ 2 ) one finds (after changing to a coordinate system that covers the horizon region) from (1.2) that if K = ∂/∂t then κ = 1 2 dh/dr, which is of the form of the negative of the gradient of the gravitational potential, evaluated on the horizon. If h = (r − r+)(r − r−)/r 2 , as in the Reissner-Nordström metric, then κ+ = (r+ − r−)/(2r 2 + ) > 0, while κ− = −(r+ − r−)/(2r 2 − ) < 0. In general, of course, the slope of h(r) must always have opposite signs at two adjacent zeros, and thus the surface gravities must have opposite signs. event horizon in an asymptotically flat spacetime (for which in fact κ is positive), the temperature is κ/(2π). We shall discuss the extension of Hawking's calculation to the case of inner horizons in the concluding section of this paper. In what follows, however, we shall frequently make reference to the formula with the understanding that T may not be a temperature measured by a physical thermometer, but rather, as we shall explain shortly, a "Gibbsian" temperature.
The occurrence of a negative κ on an inner horizon is somewhat obscured in many discussions in the literature by the fact that the surface gravity is commonly calculated by evaluating in the limit on the horizon. This formula is derivable from (1.2), but the information about the sign of κ is lost, and commonly the positive root is assumed when calculating κ from (1.4). A guaranteed correct procedure is to use the formula (1.2), working in a coordinate system that covers the horizon region.
Another situation where one encounters two horizons is when a positive cosmological constant Λ is involved and one has both a black hole event horizon and a cosmological event horizon bounding a static or stationary region [28]. A number of recent studies have pointed out that the surface gravities of the black hole horizon κ B and the cosmological event horizon κ C again have opposite signs [29][30][31]. Most have followed the procedure adopted in [28] and taken the physical temperature to be |κ| 2π (for example, see [32]). A similar situation arises in the case of the C-metrics, which contain both a black-hole horizon and an acceleration horizon. Their surface gravities are of opposite signs.
In order to assess the status of these suggestions, in this paper we shall re-examine the foundations of classical black hole thermodynamics from the viewpoint of the approach to classical thermodynamics advocated by Gibbs [33]. The central idea of this approach is that the physical properties of a substance are encoded into the shape of its Gibbs surface, i.e. the surface given by regarding the height of the surface as given by the total energy, regarded as a function of the remaining extensive variables. From this point of view, the temperature is given by the slope of the curve of energy versus entropy. To this end, we shall need explicit Christodoulou-Ruffini formulae, and a major goal of this paper is to obtain these for a variety of black hole solutions. As we shall see, it is a common feature of these examples that the "Gibbsian temperature" thus defined, while positive for black hole event horizons, is negative for inner horizons (i.e. Cauchy horizons) and for cosmological horizons. We shall return to a discussion of the physical consequences for spacetimes with two horizons in the conclusions.
Letus recall that the formalism of thermodynamics, applied to classical black holes, began with two independent discoveries: • Christodoulou's concept of reversible and irreversible transformations such that the energy E of a rotating black hole of angular momentum J and momentum P may be expressed as where the irreducible mass M irr is non-decreasing [34] • Hawking's Theorem [35,36] that the area A of the event horizon is non-decreasing.
In fact and for charged rotating Kerr-Newman black holes and dropping the momentum contribution and setting J = |J|, one has [37] the Christodoulou-Ruffini formula: (1.7) The obvious analogy of some multiple of the area of the horizon with entropy became even more striking with the discovery by Smarr [38] of an analogue of the Gibbs-Duhem relation for homogeneous substances. For Kerr-Newman black holes, this reads M = 1 4π κA + 2ΩJ + ΦQ , (1.8) where κ is the surface gravity, Ω the angular velocity and Φ the electrostatic potential of the event horizon. The analogy became almost complete with the the formulation of three laws of black hole mechanics, including the first law dM = 1 8π κdA + ΩdJ + ΦdQ , (1.9) by Bardeen, Carter and Hawking [39]. Note that the Smarr relation (1.8) follows from the first law (1.9) by differentiating the weighted homogeneity relation M (λ 2 A, λ 2 J, λQ) = λM (1. 10) with respect to λ and then setting λ = 1.
The existence of a "first law" is not in itself surprising, nor does it, in itself, imply any thermodynamic consequences. Whenever one has a problem involving varying a function subject to some constraints, and considering the value of the function at critical points, one has a formula analogous to (1.9). In the case of black hole solutions of the Einstein equations, they are known to satisfy a variational principle in which the mass is extremised keeping the horizon area, angular momenta and charges fixed (see, for example, [40][41][42]).
Similar formulae arise in the theory of rotating stars (see, for example, [43]). The study of these variations is sometimes referred to as comparative statics.
For homogeneous substances with pressure P , volume V and internal energy U , it is well known that the Gibbs-Duhem relation is equivalent to the statement that the Gibbs free energy, or thermodynamic potential, Classically, a number of arguments led to the conclusion that the laws of black hole mechanics were just analogous to the laws of thermodynamics. One argument was that as perfect absorbers, classical black holes should have vanishing temperature and hence the entropy should be infinite (cf. [44,45]). Another was based on dimensional reasoning.
In units where Boltzmann's constant is taken to be unity, entropy is dimensionless, but in classical general relativity it is not obvious how to achieve this without introducing a unit of length. The obvious guess for entropy would be some multiple of the area A, but why not some monotonically increasing function of the area? Despite these doubts it was conjectured by Bekenstein [45] that when quantum mechanics is taken into account some multiple of A l 2 p should correspond to the physical entropy of a black hole. This conjecture was subsequently confirmed at the semi-classical level by by Hawking [46,47], using quantum field theory in a curved background. Given this, one recognises the Christodoulou-Ruffini To summarise, the purpose of the present paper is to re-examine these issues systematically, based on Gibbs's geometric viewpoint of the mathematical formalism of thermodynamics [33]. This starts with a choice of pairs of extensive and intensive variables and an expression for some sort of "energy," which is regarded as a function of the extensive variables. For the black holes in asymptotically flat spacetimes that we shall consider, the energy is taken to be the ADM mass M , and the extensive variables S µ are usually taken to be S µ = (S, J, Q i , P i ) = (S, s), where S = 1 4 A and A is the area of the event horizon, J is the total angular momentum and Q i and P i are 2N electric and magnetic charges. 2

Thus we have
(1.14) The intensive variables are taken to be where T is the temperature, Ω is the angular velocity of the horizon, and Φ i and Ψ i are the electrostatic and magnetostatic potentials.
The organisation of this paper is as follows. In section 2, we review the theory of Gibbs surfaces, and the various thermodynamic metrics with which they may be equipped. Section 3 forms the core of the paper. In it, we give many new results for the thermodynamics of a wide range of asymptotically-flat black holes. We begin in subsections 3.1, 3.2 and 3.3 by reviewing how the well-known Reissner-Nordström, Kerr and Kerr-Newman black holes fit into the Gibbsian framework. Subsection 3.4 then provide a extensive discussion of the thermodynamics of families of black holes in four-dimensional STU supergravity. In particular, we give a systematic discussion of the notion of the decomposition of the system into left-handed and right-handed sectors, and their associated thermodynamics. Subsection 3.5 has analogous results for five-dimensional STU supergravity black holes. Subsections 3.6 and 3.7 give similar results for the general family of four-dimensional Einstein-Maxwell-Dilaton (EMD) black holes, and a two-field generalisation. Included in the discussion of these two-field EMD black holes, we exhibit a new area-product formula.
A rather general feature of many asymptoticaly flat black holes with two horizons is that the product of the areas of the two horizons is independent of the mass, and given in terms of conserved charges and angular momenta, which may plausibly be quantised at the quantum level. In section 4, we use this area-product formula to exhibit an intiguing S → 1/S inversion symmetry of the Christodoulou-Ruffini formulae for such black holes.
This symmetry of the Gibbs surface interchanges the positive and negative temperature branches.
In section 5 we extend our discussion to black holes that are asymptotically AdS, or black holes with positive cosmological constant. In the AdS case the situation for inner and outer horizons is broadly similar to that for the asymptotically flat case. For positive cosmological constant, the black hole event horizon continues to have positive Gibbsian temperature, but that of the cosmological horizon is negative.
In section 6, we revisit an old observation, that the entropy of the Kerr-Newman solution is a super-additive function of the extensive variables, and we its relation to Hawking's area theorem for black-hole mergers. We find that super-additivity holds also for a wide variety of the asymptotically-flat examples that we considered in section 3. However, we find that Kaluza-Klein dyonic black holes provide a counterexample, and we speculate on the reason for this.
The paper ends with conclusions and future prospects in section 7.

The Gibbs surface
In this section we shall briefly summarise those aspects of the Gibbs surface which are relevant for the latter part of the paper. If we think of (S µ , M ) as coordinates in R 3+2N then (1.14) defines a hypersurface G ⊂ R 3+2N whose co-normal is (T µ , −1). Since in our case M is a unique function of the extensive variables, the intensive variables are unique functions of the extensive variables: T µ = T µ (S ν ). The converse need not be true. If the function M (S µ ) were convex, then for fixed co-normal (T µ , −1) the plane would touch the surface at a unique point (S µ , M ) . For a smooth Gibbs surface G, convexity requires that the Hessian be positive definite and one may then define a positive definite metric called the Weinhold metric [49]. Because one of the components of the Weinhold metric (2.2) is related to the heat capacity 3 at constant J and Q i and P i , namely and neutral black holes or black holes with small charges or angular momentum have negative heat capacities, the Gibbs surface G is typically not convex and the Weinhold metric for black holes is typically Lorentzian [50].
If one defines a totally symmetric co-covariant tensor of rank three by the Riemann and Ricci tensors and the Ricci scalar of the Weinhold metric are given by all indices being raised with g µν W , the inverse of g W µν . Divergences in R are sometimes held to be a diagnostic for phase transitions.
The geometry of the Gibbs surface is essentially the geometry behind the first law of thermodynamics. As we remarked previously, this fits into a pattern that is more general than just the theory of black holes, and arises whenever one is considering a variational problem with constraints. Since this is not as widely known as it deserves to be, we shall pause to describe the general situation, and then we shall restrict attention to its application to black hole theory. Consider a real-valued function f (x) on some space X with coordinates x, subject to the n constraints that certain functions C a (x) = c a , 1 ≤ a ≤ n, where the c a are constants. Adopting the method of Lagrange multipliers, we require that Since, when pulled back to LA we have T µ dS µ = dM (S µ ), the pull-back of ω to LA vanishes, In other words, LA is a Lagrangian submanifold of R 6+3N .
One may go a step further and lift LA to R 7+2N with coordinates (P µ , S ν , M ), equipped with the contact form 14) The Gibbs function, or thermodynamic potential, G, is the total Legendre transform of the mass with respect to the extensive variables. It satisfies where Note that G is not necessarily a single-valued function of the intensive variables T µ , unless the Gibbs surface G is convex. The Hessian of the Gibbs function with respect to the intensive variables is related to the Weinhold metric by the easily-verified formula It provides a metric on the space of intensive variables.
It is important to realise that from the point of view of the symplectic and contact struc- From the point of view of the Gibbs surface G, geometrically this should really be thought of as an n-dimensional Legendrian sub-manifold of the (2n + 1)-dimensional Legendre manifold whose coordinates consist of the the total energy and the the n pairs of intensive and extensive variables . Given a choice of n coordinates chosen from these 2n variables, one may locally describe the surface in terms of the associated thermodynamic potential, and from that compute the associated Hessian metric. But globally, it is not in general true that the Gibbs surface equipped with the choice of Hessian is a single-valued non-singular graph over the n-plane spanned by the chosen set of n coordinates. It should also be remembered that although the Hessian metrics may be thought of as the pull-back to G of a flat metric on the 2n-dimensional flat hyperplane spanned by the choice n pairs of intensive and extensive variables, the signature of that flat metric depends upon that choice.
Here we review some key results on the general classes of thermodynamic metrics that were presented in [51]. Consider first the energy M = M (S µ ), which obeys the first law One can define from this the metric where T µ are viewed as functions of the S µ variables, with and ⊗ s denotes the symmetrised tensor product. In the usual parlance of general relativity we may simply write (2.19) as In view of (2.20) we have which is nothing but the Weinhold metric.
One can obtain a set of conformally-related metrics by dividing (2.18) by any one of the intensive variables T µ for µ =μ whereμ denotes the associated specific index value of the chosen intensive variable, and then constructing the thermodynamic metric ds 2 (Sμ) for the conjugate extensive variable by using the same procedure as before [51]. Thus, for example, if we chooseμ = 0, so that T is the chosen intensive variable and S its conjugate, then we rewrite (2.18) as where we have split the µ index as µ = (0, a), and then write the associated thermodynamic metric The second line was obtained by using (2.18), and the third line follows from (2.22). Thus ds 2 (S), which is the Ruppeiner metric, is conformally related by the factor −1/T to the Weinhold metric. Weinhold and Ruppeiner metrics were introduced into black hole physics in [50,52]. The literature is by now quite extensive. For a recent review see [53]. Other conformally-related metrics can be defined by dividing (2.18) by any other of the intensive variables and the repeating the analogous calculations. For example, if there is a single charge Q and potential Φ, then dividing the first law dM = T dS + ΦdQ + · · · by Φ and calculating the metric ds 2 (Q), one obtains Further thermodynamic metrics that are not merely conformally related to the Weinhold metric can be obtained by making Legendre transformations to different energy functions before implementing the above procedure [51]. For example, if one make the Legendre transform to the free energy F = M − T S, for which one has the first law then the associated thermodynamic metric will be where S and T a , which are now the intensive variables, are viewed as functions of T and S a . The metric components in ds 2 (F ) are therefore given by the Hessian of F . As observed in [51], the metric ds 2 (F ) has the property that, unlike the Weinhold or Ruppeiner metrics, its curvature is singular on the so-called Davies curve where the heat capacity diverges.
Clearly, by making different Legendre transformations, one can construct many different thermodynamic metrics, which take the form where each η µ can independently be either +1 or −1. The overall sign is of no particular importance, and so metrics related by making a complete Legendre transformation of all the intensive/extensive pairs in a given energy definition really yields an equivalent metric.
For example, the Gibbs energy G = M − T µ S µ gives the metric which is just the negative of the Weinhold metric ds 2 (M ) in (2.22).
One further observation that was emphasised in [51] is that one is not, of course, obliged when writing a thermodynamic metric to use the associated extensive variables as the coordinates. It is sometimes the case, as we shall see in later examples, that although one can calculate the thermodynamic variables in terms of the metric parameters, one cannot explicitly invert these relations. In such cases, one can always choose to use the metric parameters as the coordinates when writing the thermodynamic metrics. Geometric invariants such as the Ricci scalar of the thermodynamic metric will be the same whether written using the thermodynamic variables or the metric parameters, since one is just making a general coordinate transformation. Thus even in cases where the relations between the thermodynamic variables and metric parameters are too complicated to allow one to find an explicit Christodoulou-Ruffini formula to define the Gibbs surface, one can still study the geometrical properties of the various thermodynamic metrics.

Asymptotically Flat Black Holes
In this section, we shall illustrate the issues raised in the previous section by listing the cases of asymptotically-flat black holes for which we have explicit formulae. Whilst the formulae for the Kerr-Newman family of black holes are well known, we first review these in some detail in preparation for our discussion of much less well known black holes, such as those that occur in supergravity or Kaluza-Klein theories.

The Gibbs surface for Reissner-Nordström
The Gibbs surface G for the Reissner-Nordström solution is given by the Christodoulou- where M irr = S 4π . It is convenient to envisage (M, Q, S) as a right-handed Cartesian coordinate system with M > 0 taken vertically and −∞ < Q < ∞ and S > 0 spanning a horizontal half-plane. In (M, Q, S) coordinates the surface is part of the quadratic cone We have with M > |Q| being sub-extremal black holes. Rewriting (3.2) as the two solutions for S at fixed M and Q are given by with these corresponding to the entropies ( i.e. one quarter the area) of the outer (S + ) and inner (S − ) horizons respectively. It is straightforward to see that the temperature T = ∂M/∂S is positive on the outer horizon and negative on the inner horizon.
Equality, M = |Q|, corresponds to extreme black holes. They lie on the space curve γ extreme given by the the intersection of the two surfaces The first is a plane orthogonal to the Q plane, and the second a parabolic cylinder with generators parallel to the Q axis. The projection of γ extreme onto the Q − S plane is given by the parabola Roughly speaking, the Gibbs surface G is folded over the space curve γ extreme . Now the Weinhold metric, or equivalently the Hessian of M (S, Q), is given by Note that ∂ 2 M ∂S 2 changes sign, passing through zero, along the space curve γ Davies , given by Since the heat capacity at constant charge, C Q , is given by it also changes sign across the curve γ Davies , on which it diverges [54]. This is often taken as a sign of a phase transition. In support of this interpretation, it has been shown [55] that the single negative mode of the Lichnerowicz operator passes through zero and becomes positive as Q is increased across γ Davies .
The curve γ Davies is an example of what, in the literature on phase transitions, is often referred to as a spinodal curve, and is usually defined in terms of the vanishing of a diagonal element of the Hessian of the Gibbs function. In the present case, the Gibbs function is 11) and the Hessian is given by  The spinodal curve is thus given by Φ 2 = ± 1 3 , which, in terms of S and Q, coincides with (3.9).
The Weinhold metric may written as and hence the Gibbs surface for sub-extremal black holes has a Hessian, or equivalently a Weinhold metric, that is non-singular but Lorentzian. Moreover the Gibbs surface for nonextreme black holes is non-convex. Expressed in terms of S and the electrostatic potential the Weinhold metric becomes Note that the metric is non-singular when either Φ 2 < 1, corresponding to the outer horizon, or Φ 2 > 1, corresponding to the inner horizon. It changes signature from (−+) to (++) as Φ goes from Φ 2 < 1 to Φ 2 > 1. The heat capacity passes through infinity at Φ 2 = 1 3 . Expressed in terms of Φ and S, the temperature is given by , and so the Ruppeiner metric is given by where we have defined, for the outer horizon, The metric in the second line of (3.16) is the Milne metric on a wedge of Minkowski spacetime inside the light cone. This is made apparent by introducing new coordinates according to in terms of which the Ruppeiner metric becomes , the extremal solutions lie on the timelike geodesics t = ± arctanh π √ 2 . The heat capacity changes sign at Φ 2 = 1 3 . If Φ 2 > 1, corresponding to the inner horizon, then, if Q > 0, substituting The metric in brackets is the flat metric on Euclidean space in polar coordinates, except that The flatness of the Ruppeiner metric for Reissner-Nordström has given rise to much comment, because singularities of the Ruppeiner metric are expected to reveal the occurrence of phase transitions. However, the geometrical significance of the change in sign of the heat capacity is that for fixed charge Q, there is a maximum temperature. In fact so for given |Q| and positive T less than √ 3 8π|Q| , there are two positive values of S and hence two non-extreme black holes. By contrast, since the electrostatic potential Φ satisfies (3.14), there is a unique positive value of S and hence a unique black hole for given Q and Φ 2 < 1.
Every two-dimensional metric is conformally flat. Therefore it is not surprising that both the Weinhold and Ruppeiner metrics for Reissner-Nordström are conformally flat. It is, however, nontrivial that the Ruppeiner metric is flat. It has recently been pointed out [56] that one can also consider the Hessian of the charge Q, considered as a function of the mass and entropy, as a metric ds 2 Q . In fact ds 2 Q = −Φ −1 ds 2 W , as in (2.25). Geometrically, there is no reason to give a preference to any of the metrics ds 2 W , ds 2 R or ds 2 Q . Since T and Φ are both non-singular on the curve along which the heat capacity diverges, none of the three metrics is capable of detecting the associated "phase transition." As was shown in [51], and we reviewed in section 2.2, the thermodynamic metric (2.27) constructed from the free energy F = M − T S does exhibit a singularity on the Davies curve where the heat capacity diverges. For the Reissner-Nordström metric (2.27) is the restriction of ds 2 (F ) = −dT dS + dΦdQ to the Gibbs surface, and hence we find (3.23) A straightforward calculation shows that its Ricci scalar is given by which does indeed diverge on the Davies curve S = 3πQ 2 .

The Gibbs surface for Kerr
This is qualitatively very similar to the Reissner-Nordström case. To begin with, we shall summarise, in our notation, some results first presented by Curir [1]. One has and M (S, J) at fixed J has a minimum value when This is the extreme case and, as before, the inner horizon has a negative temperature, a point made first by Curir [1]. Explicitly one has By (3.27), the outer horizon has a positive temperature, which we label T + , and the inner horizon has a negative temperature, which we label T − . One has [1] where Ω ± = (∂M/∂J) S ± . Note that it follows from the first equation in (3.29) that Note also that M and J, which are conserved quantities defined in terms of integrals at infinity, are universal and do not carry ± labels.
In terms of S + and S − , one has, from (3.25), There is also a modified Smarr formula where the second equality follows from (3.30). This way of writing the first law of thermodynamics was employed in [57] for deriving a simple formula for holographic complexity.
These results were interpreted in [1] as indicating that the total energy of a rotating black hole may be regarded as receiving contributions from two thermodynamic systems; one associated with the outer horizon and the other with the inner horizon. The negative temperature was interpreted in terms of Ramsey's account of the thermodynamics of isolated spin systems [58].
Okamoto and Kaburaki [10] introduced the dimensionless parameter h = in their discussion of the energetics of Kerr black holes and noticed that it satisfies the quadratic equation It was initially assumed that only the solution of (3.35) satisfying 0 ≤ h ≤ 1 has physical significance. However Abramowicz [59] drew their attention to [1,2], and they realised that the other root of (3.35), which satisfies 1 ≤ h ≤ ∞ and is given by h = a M − √ M 2 −a 2 , is associated with the inner horizon [10]. Expressing the thermodynamic variables in terms of h they established (3.30) if T − is taken to be negative, and they also obtained the formula

Kerr-Newman black holes
Kerr-Newman black holes may have both electric and magnetic charges. By electricmagnetic duality invariance one may set the magnetic charge P to zero. To restore electricmagnetic duality invariance it suffices to replace Q 2 by Q 2 +P 2 in all formulae thus producing a manifestly O(2) invariant Gibbs surface.
The mass of the Kerr-Newman black hole is given by and therefore it satisfies acquiring its least value on the surface γ extreme in the three dimensional space of extensive variables given by on which the temperature vanishes. If J = 0, then (3.38) is the usual Bogomolnyi bound [60]. One also has The explicit formulae (3.37), (3.40) and (3.41) allow a lift of the Gibbs surface G to a Lagrangian submanifold L in R 6 and a Legendrian submanifold in R 7 . The entropy product law becomes where the − refers to the inner and + to outer horizon.
The temperatures and angular velocities of the two horizons are given by and one has There is a conventional first law for both horizons: and a modified Smarr formula

STU black holes
Four-dimensional black holes in string theory or M-theory can be described as solutions of N = 8 supergravity. The most general black holes are supported by just four of the 28 gauge fields, in the Cartan subalgebra of SO (8). The black holes can therefore be described just within the N = 2 STU supergravity theory, which is a consistent truncation of the N = 8 theory whose bosonic sector comprises the metric, the four gauge fields, and six scalar fields. Black holes of the STU model are parameterised by mass M , angular momentum J and four electric Q i (i = 1, 2, 3, 4) and four magnetic charges P i (i = 1, 2, 3, 4 We shall follow the usual conventions for STU supergravity, in which the normalisation of the gauge fields F (i) is such that if the scalar fields are turned off, the Lagrangian will for a presentation of the bosonic sector of the STU supergravity Lagrangian). This contrasts with the conventional , in Gaussian units, which we use when describing the pure Einstein-Maxwell theory. Since this means that the charge normalisation conventions will be different in the two cases, we shall briefly summarise our definitions here. If we consider the Lagrangian one can derive by considering variations of the associated Hamiltonian that black holes will obey the first law where κ is the surface gravity, Φ is the potential difference between the horizon and infinity (with the potential being equal to ξ µ A µ , where ξ µ is the future-directed Killing vector that is null on the horizon and is normalised such that ξ µ ξ µ → −1 at infinity). The electric charge Q is given by while in STU supergravity we shall have (neglecting the scalar fields for simplicity 4 ) The black hole solutions have two horizons, with the the product of the horizon entropies quantised: where ∆ is the Cayley hyperdeterminant ∆(Q i , P i ): Note that eqn (3.52) has previously appeared in the literature without the absolute value symbol (for example, in [61]). We have written (3.52) with an absolute value sign since ∆, and hence ∆ + J 2 , can be negative; for example for a static Kaluza-Klein dyonic black hole.
(In [61] it was proposed that S − is negative when ∆ + J 2 < 0, but this would contradict the fact that, for example, the area of the inner horizon of the static Kaluza-Klein dyonic black hole is positive.) It should be noted that if J vanishes and ∆ = 0, then S − will vanish also. In this case there is no non-singular inner horizon.
The entropy formulae (3.52) can be cast in the form where F is another complicated expression that is a function of M , Q i and P i only [61].
Note that it follows from (3.54) that S + ≥ S − . Unlike [61], we have put an absolute value sign around (S L − S R ) in the expression for S − , since, for the reasons discussed above, there can be circumstances where S L < S R , but S − should be non-negative. Note that F + ∆ is always non-negative, and F − J 2 is non-negative provided that the black hole is 4 In general, including the scalar fields, and writing the Lagrangian as a 4-form, we shall have L = The electric charges can be written as (Here the variational derivative is defined by δX = (δX/δF ) ∧ δF . For example if X = u * F ∧ F + v F ∧ F then δX/δF = 2u * F + 2v F .) The magnetic charges are given by not over-rotating [61]. The quantities S L and S R are both non-negative. In the extremal limit F − J 2 = 0, one gets the extremal value for the entropy S + = S − = 2π |∆|. This was seen for the BPS solutions (F = 0 and J 2 = 0) in [13].
Note from (3.55) that while the right-moving entropy S R is a function of all the extensive variables (M, Q i , P i , J), the left-moving entropy S L is a function of (M, Q i , P i ) but not J [61]. This was noted previously in the special case of the four-charge black holes characterised by (M, Q i , J) in [11,62]. The expressions (3.55) may in principle be inverted to give two different Christodoulou-Ruffini formulae: The structure (3.55) ensures that the two entropies S + and S − are solutions of the quadratic equation where Σ = S L + S R + |S L − S R |, and we employed (3.54), (3.55) and (3.52). Note that Since S + ≥ S − , the final expression in (3.58) is non-negative for S = S + , and non-positive is independent of whether one takes S = S + or S = S − , it then follows that In particular, this implies that T + and T − must have opposite signs.
As well as considering the left-moving and right-moving entropies S L and S R , one can also introduce left-moving and right-moving temperatures T L and T R , defined by [15] 1 These definitions are motivated by the fact that when one calculates scattering amplitudes for test fields propagating in the black-hole background, one finds that they factorise into the product of thermal Boltzmann factors for the temperatures T L and T R respectively [15].
Using (3.59), together with the expressiona for S + and S − in terms of S L and S R in (3.54), it follows from (3.60) that for the two cases that we described previously. From its definition, T R is obviously nonnegative since T + ≥ 0 and T − ≤ 0. It is then evident from (3.61) that T L is non-negative also, since we already know that S L and S R are non-negative.
We can also derive, from and using either (3.57) or else simply writing S + and S − in terms of S L and |J 2 + ∆| by using (3.52), that in the two cases S L ≥ S R and S L ≤ S R we have Note that when S L < S R , i.e. when J 2 + ∆ < 0, the angular velocities of the inner and outer horizons are opposite. Note also that the two cases in (3.63) can be expressed in the single universal formula For now, we shall focus for simplicity on the regime where S L ≥ S R , i.e. (J 2 + ∆) ≥ 0.
Let us first consider processes where dJ = 0 and dQ i = dP i = 0. From the definitions of T L , T R , S L and S R given in (3.54) and (3.60), it straightforward to see from the first laws on the outer and inner horizons that we must have In other words, the left-moving and right-moving sectors contribute equally to the mass of the black hole. (This was observed in the case of Kerr-Newman black holes in [23,24].) Dividing the first laws (3.66) by T ± respectively and then taking the plus and minus combinations, one finds that these match with (3.65) provided that we define the left-moving and right-moving quantities as and so we have the first laws for the left-moving and right-moving sectors.
In a similar fashion, we can then see that the Smarr relations on the outer and inner horizons imply the Smarr relations for the left-moving and right-moving sectors.
It should be noted that, from (3.63) and (3.68), the left-moving angular velocity is in fact zero: If we now turn to the regime where S L < S R , we find that the roles of S L and S R are exchanged in both the first laws and the Smarr relations for the left-moving and right-moving sectors, so that we have

Four-charge STU black holes
The prospects for obtaining an explicit Christodoulou-Ruffini formulae for the general 8charge black hole solutions are not good. The main problem is the F -invariant that appears in the expressions for S L and S R in eqn (3.55), whose evaluation in terms of physical charges and mass appears to be quite intractable [63]. In order to obtain more explicit, concrete expressions, we shall now focus on the specialisation to black-hole solutions carrying just four electric charges, which were found in [11].
These black holes are parameterised in terms of the non-extremality parameter m ≥ 0 (Kerr mass parameter), the "bare" angular momentum a (Kerr rotation parameter) and four boost parameters δ i ≥ 0 (i = 1, 2, 3, 4) [11] (see also [64] for compact expressions for the metric and the other fields). In terms of these, the physical mass, charges and angular momentum are given by The black hole entropies, associated with the inner and the outer horizon, are given by [11,16]: The temperatures T ± , related to surface gravities κ ± by T ± = κ ± 2π , and angular velocities Ω ± , which are associated with the inner and out horizon respectively, are given by [16]: ) where Note that T − is negative. 6 From the above expressions one also finds It can easily be verified that the entropies S ± , temperatures T ± and angular veocities Ω ± satisfy equation (3.59) and the S L ≥ S R equations in (3.63).
The entropies and the inverses of the surface gravities, associated with the outer and inner horizons, have a suggestive form in terms of the left-moving and right-moving entropy and inverse temperature excitations of a weakly coupled 2-dimensional conformal field theory (2D CFT), given in [16]: 1 1 Note that these solutions with four electric charges have ∆ ≥ 0, as can be seen from (3.53), and so they have S L ≥ S R , as is evident from (3.82). In this suggestive form the central charges C L,R of the left-moving and right-moving sector of the the 2D CFT, related to S L,R and T L,R via Cardy relation S L = π 2 3 C L T L and S R = π 2 3 C R T R , respectively, turn out to be the same and equal to: Again the product of outer and inner horizon entropies is quantized in terms of J and Q i (i = 1, 2, 3, 4) only [18]: which agrees with the result for Kerr-Newman black hole after equating The main challenge here is to obtain the formulae M = M (S, J, Q i ) and S = S(M, J, Q i ).
As an initial step, we observe the solutions for S ± , due to relation (3.82), satisfy a quadratic equation: and From (3.88) we obtain: Furthermore, employing (3.89) and (3.90) we obtain: which leads to the explicit expression for the temperature: 93) and angular velocity: The technical difficulty in obtaining an explicit Christodoulou-Ruffini mass expression is due to the fact that an explicit expression for S L in terms of M and Q i is cumbersome, in general. However, we succeeded in the following special cases.

Pairwise-equal charges
The four-charge black-hole solutions simplify considerably in the special case of pair-wise equal charges (see, for example, [64]) Q 1 = Q 3 and Q 2 = Q 4 where (3.88) can be solved explicitly for M: Furthermore (3.95) and (3.86) implies: For Q 2 = 0 the result reduces to the example of rotating dilatonic black hole with the dilaton coupling a = 1. 7 The result reduces to the Kerr-Newman (or Reissner-Nordström) black hole expression when Q 1 = Q 2 = 1 4 Q. It becomes straightforward that the differentiation of (3.95) with respect to S ± (with J and Q 1,2 fixed), produces the expected expressions for T ± , including the sign.

Three equal non-zero charges
It turns out that for the example of three equal non-zero charges, i.e. Q 1 = Q 2 = Q 3 = q and Q 4 = 0, which corresponds to the rotating dilatonic black hole with the dilaton coupling a = 1 √ 3 , one can again obtain an explicit expression for the the Christodoulou-Ruffini mass: (3.97) 7 Note, however, that when the black hole is rotating, an axion in the STU supergravity is also turned on when Q1 and/or Q2 is non-zero (except in the case Q1 = Q2).
(As in the pairwise-equal charge case above, here too an axion is also turned on if the black hole is rotating.)

One non-zero charge
We also note in the case of only one non-zero charge (say, Q 1 = q = 1 4 m sinh 2δ), which corresponds to the rotating dilatonic black hole with the dilaton coupling a = √ 3, the Christodoulou-Ruffini mass can be expressed in the following form: , and cosh δ is a solution of the cubic equation

Dyonic Kaluza-Klein black hole
In all the explicit STU supergravity black holes we have discussed so far, each of the four field strengths carries a charge of a single complexion (which could be pure electric or pure magnetic). The most general possibility is where each field strength carries independent electric and magnetic charges, as described in the general 8-charge case that was constructed by Chow and Compère. Although explicit, these general solutions are rather unwieldy. Here, we discuss a much simpler case, which is still rather non-trivial, and that goes beyond what we have explicitly presented so far. We consider the case where just one of the four field strengths is non-vanishing, but it carries independent electric and magnetic charges. For simplicity we shall restrict attention to the case of static black holes. The Lagrangian (in the normalisation we are using for the STU supergravities) is given by 8 and a convenient way [65] to present the static dyonic black hole solutions is where m, β 1 and β 2 are constants that parameterise the physical mass M , electric charge Q and magnetic charge P , with A necessary condition for regularity of the black hole is 0 ≤ β i ≤ 1. The entropy of the outer horizon, located at r = 2µ, is given by whilst the entropy of the inner horizon, located at r = 0, is given by The product of the entropies on the outer and inner horizons is given by Note that S − vanishes if Q or P vanishes. Note also that the dyonic black hole is an example where the invariant ∆, defined in (3.53), is negative. Of course since the solutions we are considering here are static, (J 2 + ∆) is negative too, and so we are in the regime where S L < S R for these black holes, and in fact we have One can straightforwardly calculate the temperatures on the oouter and inner horizons, finding as usual that the temperature T + is positive and T − is negative. The left-moving and right-moving temperatures, defined by (3.60), then turn out to be These are both non-negative.
A special case is when the black hole is extremal, which is achieved in this parameterisation by taking a limit in which m goes to zero and the β i go to 1. The result is that in the extremal case M ext = Q

Five-dimensional STU supergravity
Here, we consider black hole solutions in five-dimensional STU supergravity. General solutions with mass M , two angular momenta J φ and J ψ , and three charges Q i were constructed in [12] by employing solution generating techniques. We use principally the conventions of [15], except that we shall use the labels ↑ and ↓ to denote the sum and difference combinations of the angular momenta and angular velocities associated with the φ and ψ azimuthal coordinates, reserving L and R to denote the combinations of inner and outer horizon quantities, analogous to the definitions used previously for the four-dimensional STU black holes. The physical mass, charges and angular momenta are given by [15] Here the five-dimensional Newton constant is taken to be G 5 = π 4 . We shall, without loss of generality, take the rotation parameters l 1 and l 2 and the charge boost parameters δ i to be non-negative in what follows.
These black holes have many analogous properties to those of the four-dimensional STU black holes, except, of course, that they can carry only electric charges but not magnetic.
In particular, they have two horizons, with the inner and outer horizon entropies expressed as [15]: The product of the inner and outer horizon entropies is again quantised as: Note that as in the case of the four-dimensional STU black holes, here it would in general be necessary to use an absolute value in the expression for S − in (3.112), and on the righthand side of (3.115), since S − must be non-negative while S L and S R , which are both non-negative, could obey either S L > S R or S L < S R depending on the relative values of the charge and angular momentum parameters. However, our non-negativity assumptions stated above for the charge and rotation parameters imply that in fact S L ≥ S R in this case, and so we can omit the absolute value in the expression for S − , as we have done in (3.112), and in (3.115).
From the above expressions it follows that S (either S + or S − ) again obeys a quadratic equation, Furthermore one can analogously derive the general result that T + and T − have opposite signs, with: and similarly Ω ↑ where Ω ↑ ± = 1 2 (Ω φ ± + Ω ψ ± ) and Ω ↓ ± = 1 2 (Ω φ ± − Ω ψ ± ). (The relative signs between the terms in these two equations in (3.118) are the opposite of those given in [15], because in that paper κ − was taken to be positive.) The black holes obey the usual first laws on the outer and inner horizons: As in the four-dimensional case, the calculation of scattering amplitudes in the black-hole background shows that they factorise into left and right sectors with Boltzman factors corresponding to temperatures T L and T R given by (3.60) [15]. Together with the normalisation of S L and S R , such that S + = S L + S R in accordance with the interpretation of the entropy as the sum of left-moving and right-moving contributions, one can then establish by rewriting the first laws dM = T ± dS ± + · · · in terms of left and righ-moving quantities that 1 2 dM = T L dS L + · · · and 1 2 dM = T R dS R + · · · , and so each of the sectors contributes one half the total mass of the black hole. Matching the first laws for arbitrary variations of the parameters then allows one to read off the appropriate definitions of the left-moving and right-moving angular momenta and electric potentials. Thus one finds the first laws In view of the relations (3.118), one finds Thus we see that the angular momentum J ↑ and the associated angular velocity Ω ↑ enters only in the right-moving first law and in S R , while the angular momentum J ↓ and associated angular velocity Ω ↓ enters only in the left-moving first law and in S L . Note that as in four dimensions, T L and T R are both non-negative.
The Smarr formulae for the left-moving and right-moving sectors agree with the ones derived in [15]: The expression for the Christodoulou-Ruffini formula in terms solely of the conserved charges, angular momenta, mass and entropy are too cumbersome to present explicitly.
Even in the case of three equal charges, the mass is determined by a cubic equation.

Einstein-Maxwell-Dilaton black holes
There exists a more general class of black holes in the theory of Einstein-Maxwell gravity with an additional dilatonic scalar field, which is coupled to the Maxwell field with a dimensionless coupling constant a, with the Lagrangian The electrically-charged black-hole solution can be written as [66][67][68] The relevant thermodynamic quantities for these black holes in this theory are given by where r + is the radius of the outer horizon, and r − is a singular surface unless a = 0. Since by assumption r + ≥ r − , it follows that This is consistent with the BPS bound derived in [69] using "fake supersymmetry." The Smarr relations continue to hold and the Gibbs free energy is again given by The coordinates {r + , r − } are now related to the coordinates {T, Φ} by and Thus the Gibbs energy as a function of {T, Φ} is given by As discussed in section 2.2, the Ricci scalar of the Helmholtz free energy metric ds 2 (F ) = −dS dT + dΦ dQ will be singular on the Davies curve where the heat capacity at constant charge changes sign. It is easiest to use r + and r − as the coordinate variables in this calculation, which gives (3.133) Thus the Davies curve is given by (3.135) Since we must have r − < r + , a solution for (3.134) exists only for a 2 < 1. The spinodal curve thus projects down to the parabola in the S − Q plane given by From (3.127), one can in general solve for r + and r − in terms of M and Q, obtaining If a 2 > 0 the entropy vanishes at extremality, namely r + = r − and hence |Q| = √ 1 + a 2 M .
Then r = r + = r − is a point-like singularity and there is no inner horizon. One can also, in general, express the entropy in terms of r + and Q, using (3.139) Particular cases include the following, which also arise as special cases of STU Black holes: • a = 0 is the Reissner-Nordström case.
• a 2 = 1 3 is a reduction of Einstein-Maxwell in 5 dimensions.
• a 2 = 1 is the so-called string case. We have The spinodal curve coincides with the Q-axis and the Gibbs surface is nowhere convex. It is a hyperbolic paraboloid for which the Ruppeiner metric is flat [70]. The temperature is given by and is always positive. It goes to a non-vanishing value at extremality. The heat capacity at constant charge is given by and is always negative, and is also non-vanishing at extremality [67].

Two-field dilatonic black holes
Here we review a class of theories [71] which are similar to the Einstein-Maxwell-Dilaton (EMD) theory of the previous subsection, but with two field strengths rather than just one.
The Lagrangian, in an arbitrary dimension D, is given by (3.143) The advantage of considering this extension of EMD theory is that by choosing the coupling constants a 1 and a 2 appropriately, we can find general classes of static black hole solutions with two horizons, and one can study the thermodynamic properties at both the outer and inner horizon.
If we turn on both the gauge fields A i independently, the theory for general (a 1 , a 2 ) does not admit explicit black hole solutions. We shall determine the condition on (a 1 , a 2 ) so that the system will give such explicit solutions. It is advantageous for later purpose that we reparameterize these dilaton coupling constants as (3.144) (Note that N 1 and N 2 are not necessarily integers.) For the a i to be real, we must have Here we shall consider the case where a 1 and a 2 obey the constraint which implies the identities With a 1 and a 2 obeying (3.146), one can find black hole solutions, given by [71] where we are using the standard notation where s i = sinh δ i and c i = cosh δ i . The mass and charges are given by where ω D−2 is the volume of the unit (D − 2)-sphere. The outer horizon is located at 3) , and the entropy is given by The inner horizon is located at r = 0, and we have Multiplying the two entropies gives the product formula Thus the entropy product is independent of the mass.
There exists an extremal limit in which we send µ → 0 while keeping the charges Q i non-vanishing. In this limit, the inner and outer horizons coalesce and the near-horizon geometry becomes AdS D−2 × S 2 . The mass now depend only on the charges, and is given by It is useful to define and then we have Some specific examples are as follows: and then Case 2: D = 4, N 1 = 1, N 2 = 3: These cases lie, in general, outside the realm of supergravity theories. We have Entropy super-additivity is difficult to prove in general, but we can at least look at the case of extremal black holes, for which It seems that super-additivity will be satisfied if N 1 + N 2 ≥ 2, and in fact, from (3.147), we have N 1 + N 2 > in all dimensions.

Entropy Product and Inversion Laws
It is well known from many examples that if a black hole has two horizons then the product of the areas, or equivalently entropies, of these horizons is equal to an expression written purely in terms of the conserved charges and angular momenta [18,20]. Thus we may write where Q represents the complete set of charges carried by the black hole, and J represents the set of angular momenta. (Generalisations arise also if there are more than two horizons or "pseudo-horizons" (see, for example, [18]. One important consequence of the inversion symmetry of the Christodoulou-Ruffini relation M = M (S, Q, J) is that the relation S + T + + S − T − = 0, seen, for example, for the STU black holes in (3.59), is true quite generally. Since the temperature is given by ∂M/∂S at fixed Q and J we have where K = K(Q, J) is the numerator in the inversion formula (4.4). Taking S = S + we therefore have S ′ = S − , and so we find from (4.5) that whenever there is an entropy-product rule of the form (4.1) and the related inversion symmetry under (4.4).

Asymptotically AdS and dS Black Holes
In this section we shall extend the previous discussion to the case of a non-vanishing cosmological constant. If the cosmological constant is negative, the situation is similar to the case when it vanishes. However, if the cosmological constant is positive a new feature arises, namely the occurrence of an additional "cosmological" horizon outside the black hole event horizon. Typically the surface gravity at the cosmological horizon is negative.

Kottler
Either we regard Λ as a fixed constant or as an intensive variable which may be varied, in which case we obtain an analogy with a gas with positive pressure In the first case we should think of the Abbott-Deser mass M as the total energy. In the second case, we should instead think of it as the total enthalpy [72,73]. In both cases we have and in both cases and the heat capacity at constant pressure is given by We now consider the two cases where Λ < 0 and Λ > 0.
The temperature T is a positive, monotonic-increasing function of entropy S at fixed pressure P . The isobaric curve in the S − M plane has a point of inflection at which the heat capacity changes sign when where the slope, and hence the temperature, has a minimum value; It follows that for fixed negative Λ there are no black holes with temperatures less than T min . For temperatures above T min there are two black holes, one with a mass smaller than This is the location where the heat capacity diverges. It is connected with the Hawking-Page phase transition [74,75]. There is actually a region of masses M HP > M > M cr where the AdS 4 space is entropically favoured; however the black hole still has a positive heat capacity. As with the Reissner-Nordström black hole, it has been shown that the sign of the lowest eigenvalue of the Lichnerowicz operator changes sign as the heat capacity changes sign [76].

Λ > 0
We have a negative pressure, P < 0. If M is assumed positive we have two horizons, a black hole horizon with and positive temperature T = ∂M/∂S, and a cosmological horizon with for which T = ∂M/∂S < 0, and hence the temperature is negative. The heat capacity is therefore always negative. The temperature vanishes when the two horizons coincide, that is if S π = Λ , (5.10) at which the mass has a maximum of In summary, we have two horizons; a black hole horizon and a cosmological horizon.
The entropy of the former is smaller then or equal to the entropy of the latter. It seems most appropriate to regard M as the enthalpy. In this case the black hole horizon has positive temperature and the cosmological horizon has negative temperature. This differs from the usual interpretation in which both temperatures are taken to be positive. In effect one takes T C = |κ C | 2π where κ C , where κ C is the surface gravity of the event horizon [28][29][30][31]. However, even if one follows the conventional interpretation it should be borne in mind that it is not an equilibrium system and there is no period in imaginary time which would produce an everywhere non-singular gravitational instanton, except when the black hole is absent as in [28,77].

Λ < 0
If r = S π is the radius in the area coordinate, we have where Λ 3 = −g 2 . using the fact that ∂ ∂S = 1 2πr ∂ ∂r (5.13) one finds that 1 − Q 2 r 2 + 3g 2 r 2 (5.14) and thus T vanishes at r = r extreme where If we take the limit that Q 2 → 0 we obtain the spinodal curve of the the Hawking-Page phase transition [74] and if we take the limit g 2 → 0 we obtain the spinodal curve of the Davies phase transition [54]. The two curves meet at the critical point 6|gQ| = 1.

Λ > 0
horizon respectively. From the Gibbsian point of view one has T = κ 2π and therefore Because |T 1 |=T 2 we obtain a gravitational instanton by setting t = iτ and identifying τ modulo β = 1 T 2 [79] . The sign used for the period appears to have no geometrical significance and proceeding in the standard way one may argue that the two horizons are in equilibrium with respect to the exchange of thermal Hawking quanta.
It was also argued that if |κ 3 | ≥ |κ 1 , then the Cauchy horizon should be stable.

Kerr-Newman-de Sitter black holes
From [84] we take the formula Writing Λ = −3g 2 , the formula takes the form For Λ = 0 the result reduces to that of the Kerr-Newman black hole.

Pairwise-equal charge anti-de Sitter black hole
These solutions were obtained in [64], and they are special cases of solutions in the gauged STU supergravity model. (Those are also solutions of maximally supersymmetric fourdimensional theory, which is a consistent truncation of a Kaluza-Klein compactified elevendimensional supergravity on S 7 .) The theory is specified by mass M , angular momentum J, two charges, i.e., equating Q 1 = Q 3 and Q 2 = Q 4 , and cosmological constant Λ = −3g 2 .
In [64] the solution was parameterised by the non-extremality parameter m, rotational parameter a, two boost parameters δ 1,2 and g 2 . The thermodynamic quantities are of the following form: where s i = sinh δ i , c i = cosh δ i (i = 1, 2). and Ξ = 1 − g 2 a 2 . The entropy is of the form: where r i = r + + ms 2 i (i = 1, 2) and r + is a location of a horizon, which is a solution of the equation: r 2 − 2mr + a 2 + g 2 r 1 r 2 (r 1 r 2 + a 2 ) = 0 . (5.32) Manipulation of the horizon equation, along with the expressions for the M , J, Q i and S, allows one to derive the following explicit Christodoulou-Ruffini mass: (5.33)

Wu black hole
The Wu black hole [85] is 5D, three charge rotating solution with negative cosmological constant (∝ g 2 ). Employing expressions from [86] for a product of the entropy and temperature of this black hole, associated with all three horizons we obtain the following interesting expression: where Here ξ a = 1 − g 2 a 2 , ξ b = 1 − g 2 b 2 and u i is the root of the horizon equation X = g 2 (u − u 1 )(u − u 2 )(u − u 3 ). Note that as g 2 → 0, u 3 → −1/g 2 → −∞, and in this case the above equation reduces to the standard equation T 1 S 1 + T 2 S 2 = 0.

Entropy and Super-Additivity
The thermodynamics of equilibrium systems with a substantial contribution to the total energy from their gravitational self energy differs significantly from that of ordinary substances encountered in the laboratory. This is because of the long range nature of the Newtonian gravitational force, which cannot be screened. As a consequence the total entropy S of a gravitating system need not be proportional to the total energy M . A consequence of this is that negative heat capacities are possible, and indeed these have long been encountered in the theory of stellar structure [87].
In the case of black holes, the long range nature of gravitational interaction expresses itself in the fact that while the individual extensive variables may be added, they do not not necessarily scale. Even if they do, they do not scale with the same power as the total energy M . In the case of ungauged supergravity black holes, the scaling behaviour is guaranteed, but the fact that the scaling behaviour is not homogeneous, that is, not the same for all extensive variables, leads to a modification of the standard form of the Gibbs-Duhem relation for ordinary homogeneous substances where G is the Gibbs free energy, V the volume and P the pressure. By contrast, for black holes the Smarr relation (2.14) gives rise to the Gibbs function (2.16).
The requirement of homogeneous scaling plays such an important role in the thermodynamics of ordinary substances that it has been been suggested that it be called the Fourth Law of Thermodynamics [88,89]. It certainly fails for systems with significant self-gravitation and, a fortiori, for black holes. In fact if the matter sector is sufficiently non-linear such as in Einstein's theory coupled to non-linear electrodynamics, even the property of weighted homogeneity ceases to hold. 10 As a consequence, while the first law of black hole thermodynamics holds there is no analogue of a Smarr formula [90].
In the thermodynamics of ordinary substances it is usually assumed that the total energy Now if the extensive quantities scale in a uniform fashion, the property of concavity is equivalent to that of super-additivity, 12 but not necessarily if uniform scaling ceases to hold [91][92][93][94]. Remarkably, it was shown long ago in a little noticed paper by Tranah and Landsberg [93] 13 that while concavity fails for the entropy of Kerr-Newman black holes, 10 A function f (x1, x2, . . . , xn) of n variables is said to be weighted homogeneous of weights w1,w2, . . . , wn if f (λ w 1 x1, , λ w 2 x2, . . . , λ wn xn) = λf (x1, x2, . . . , xn). If wi = 1 for all i, the function is said to be homogeneous of weight one. The Fourth Law is the statement that all extensive variables have weight one and thus all intensive variables have weight zero.
concave if ≤ is changed to ≥. Subject to suitable differentiability this is equivalent to negative (positive) definiteness of the Hessian ∂ 2 f ∂x i ∂x j . In other words, if M is the total energy then the graph of the Gibbs surface along a straight line joining two equilibrium states x1 and x2 never lies above the straight line joining these points on the Gibbs surface.
12 A function f (x) is super-additive if f (x1 + x2) ≥ f (x1) + f (x2) and sub-additive if we replace ≥ by ≤. 13 Apparently not accessible on-line. The only paper we know of that has followed up on this is [8].
super-additivity remains true. In other words The super-additivity inequality (6.2) is related to Hawking's area theorem [35,36]. If two black holes of areas A 1 and A 2 can merge to form a single black hole of area A 3 , then, subject to the assumption of cosmic censorship, If the angular momentum and charge of the final black hole are equal to the sums of the angular momenta and charges of the initial black holes, one has in addition where M 3 , the mass of the black hole final state after the merger, obeys since energy will be lost by gravitational radiation. It follows from the first law that at fixed charge and angular momentum, dM = T dS and so provided that the temperature is positive, The assumption that Q 3 = Q 1 + Q 2 is reasonable for theories like Einstein-Maxwell or ungauged supergravity, where there are no particles that carry charge. The assumption that J 3 = J 1 + J 2 , however, is less reasonable, because both electromagnetic and gravitational waves can carry angular momentum.
In the following subsections we shall obtain generalisations of the Kerr-Newman superadditivity result of Tranah and Landsberg for various more complicated black hole solutions.
We also obtain a counter-example in the case of dyonic Kaluza-Klein black holes.

STU black holes with pairwise-equal charges
From the formula expressing M in terms of S, Q 1 , Q 2 and J for pairwise-equal charged STU black holes, we have For regular black holes we must have X ≥ 0 and hence Y ≥ 4Q 2 1 Q 2 2 + 16J 2 , thus implying Without loss of generality, we shall assume Q 1 , Q 2 and J are all non-negative. Note that we also have the weaker inequality which we shall use frequently in the following.
We wish to check whether the entropy of these pairwise-equal charged black holes obey the super-additivity inequality With analogous definitions for the quantities X and Y , proving super-additivity requires proving that We first note that the Y functions are non-negative, and that they obey Thus, if we can show that then the super-additivity inequality (6.10) will be established. To prove this, we first note that is can be re-expressed as We now observe that the following identity holds: where we have defined (6.17) We can use (6.16) to substitute for √ X √ X ′ in (6.15), thus yielding where we have defined The inequality (6.9) implies M ≥ Q + and M ′ ≥ Q ′ + , and a fortiori M ≥ |Q − | and M ′ ≥ |Q ′ − | (recall that we are taking all charges to be non-negative). Since P , defined in (6.16), is manifestly non-negative it follows from (6.18) that the left-hand side must be nonnegative, and hence the required inequality (6.14) is satisfied. Thus we have proven that the super-additivity property (6.10) is indeed obeyed by the entropy of the pairwise-equal charged black holes of STU supergravity.

STU black holes with three equal non-zero charges
One can also show analytically that the super-additivity property of the entropy is true for the case of STU black holes with three equal non-zero charges, say, Q 1 = Q 2 = Q 3 = q, with Q 4 = 0. In this case S = π(Y + √ X) with: where and It is straightforward to show that where w = q √ 2M and w ′ = q ′ √ 2M ′ . The second inequality in (6.23) is true for any value of {w, w ′ } ≤ 1. This result implies It is now straightforward to show that thus proving the super-additivity of the entropy in this case as well.
An analytic proof of the super-additivity of the entropy for the case of one non-zero charge follows analogous steps.
While a numerical analysis indicates that the super-additivity is true for the STU black holes with four arbitrary electric charges, it would be interesting to prove this result analytically.

Dyonic Reissner-Nordström
In the explicit examples we have studied so far, the black hole is supported by one or more field strengths that each carry a single complexion of field (pure electric charge, or instead and equivalently, one could consider pure magnetic charge). By contrast, in the next subsection we shall see that in the case of STU black holes where only a single field strength is non-zero, the dyonic black holes have an entropy that violates the super-additivity property.
The Einstein-Maxwell Lagrangian L = √ −g(R − F 2 ) admits static dyonic black hole solutions given by with mass M , electric charge Q and magnetic charge P . To have a black hole, these quantities must obey the inequality with extremality being attained when the inequality is saturated. The entropy is given by where, as usual, we assume, without loss of generality, that the charges are all non-negative.
Substituting (6.28) into this, we see that super-additivity is satisfied if First, we note that the argument of the first square root is non-negative, since, after using (6.27) for the unprimed and primed quantities we have and since the non-negativity is proven.
Returning to the inequality (6.30) that we wish to establish, we see that the terms 4M M ′ −2QQ ′ −2P P ′ are themselves certainly non-negative, since 2M M ′ −2QQ ′ −2P P ′ ≥ 0 as we just demonstrated. The inequality is therefore established if we can show that together with the analogous expression with the primes and unprimed variables exchanged.
The expression in parentheses is non-negative if is non-negative. After using (6.27) again we see that (6.34) is greater than or equal to with We shall characterise the ratio P/Q ′ by means of a constant x, such that We therefore have where the primed black hole defined above has metric parameters m and β 1 , with β 2 = 0.
This means that , (6.40) the entropy is given by and from (6.37) β 1 is given in terms of x by 14 Strictly speaking, the extremal configuration (P, 0, P ) is not a black hole, but rather a naked singularity.
However, one can make an infinitesimal deformation away from extremality, to a configuration with parameters (P + δ, 0, P ), and this will describe a genuine black hole. The results that we shall derive here, including the bound (6.46) on P versus Q ′ for obtaining violations of entropy super-additivity, are thus valid. and so super-additivity does not hold in this region of the parameter space.
When x becomes larger, we find from numerical analysis that the ratio S tot /(S + S ′ ), which equals 2 in the limit as x goes to zero, falls monotonically. The ratio reaches unity when S ′ = S tot , which implies (6.44) Substituting into (6.42), we find that this occurs when β 1 = y 3 and y is the single real root of the 9th-order polynomial 17y 9 − 12y 8 + 42y 7 − 80y 6 + 39y 5 − 48y 4 + 54y 3 − 12y 2 + 9y − 8 = 0 . It is straightforward to show from the formulae in section 3.4.5 that for the individual black holes that carry purely electric or purely magnetic charge, one has One can then use (6.47), together with (6.48), to solve for M ′ , and hence one can express Y ≡ S tot − S − S ′ , where S tot = 8πP tot Q tot , as a function of M , P and Q ′ . One can then explore the regions in the space of these parameters for which Y is negative, signifying a violation of entropy super-additivity.
Of course, by continuity we expect that super-additivity violations will occur at least in some neighbourhood of the region found above when all the masses and charges are allowed to be adjusted. In other words, there will also be super-additivity violations if we consider cases where all three black holes are non-extremal, for appropriate ranges of the various masses and charges.
In our earlier remarks relating super-additivity to the Hawking area theorem, we assumed not only cosmic censorship but also that the coalescence of the two black holes was allowed physically. In the case of dyons, it should be recalled that they carry angular momentum, and moreover it is not localised within the event horizon. This, as suggested in [95], may lead to restrictions on what coalescences are allowed, and thus the non-super additivity of the entropy in this counter-example need not imply any conflict with Hawking's area theorem. This is an interesting problem worthy of further study.

Conclusions and Future Prospects
We shall turn in this section to a consideration of the significance of negative surface gravities, and negative Gibbsian temperatures. We shall begin by recalling the most physically convincing argument that Schwarzschild black holes have a temperature, and hence entropy. This was given by Hawking [46,47], who coupled a collapsing black-hole metric in an asymptotically-flat spacetime to a quantum field, and showed that if the quantum field was initially in its vacuum state, then at late times it would emit particles with a thermal spectrum and temperature given by (1.3 The situation with two event horizons is more complicated. In order to discuss quantum fields between the horizons, one needs to specify a notion of positive frequency on each past horizon. If the region is static, and one interprets positive frequency as being with respect to a local Kruskal coordinate on the horizons, the resulting quantum state will describe thermal radiation entering the static region at temperatures given by 1 2π |κ ± |. This is not a state in thermal equilibrium. If the region between the two horizons is not static, as for example in the Reissner-Nordström solution, one may define a similar state which would also not be in thermal equilibrium. If, on the other hand, one considers the static region behind the inner horizon in the Reissner-Nordström, one needs to specify boundary consitions on the singularity at r = 0. If one chose the notation of positive frequency on the past inner horizon, then whatever boundary conditions were chosen on the singularity, the quantum state would contain radiation coming from the inner horizon with a temperature 1 2π |κ ± |. Thus if we adopt this procedure, we see in all cases that the temperature we associate with particles coming from the horizons is given by the absolute value of the surface gravity, divided by 2π.
An alternative way of establishing the temperature and entropy of an asymptoticallyflat black hole is to follow the procedure of [77,96], in which one analytically continues the metric to imaginary time, and discovers that the metric is periodic in imaginary time with a period given by 2π/|κ|, which is what one expects for a state in thermal equilibrium at temperature 1 2π |κ ± |. Of course, the period itself can have either sign, but the quantum state would not necessarily exist if one chose a negative sign for the temperature. This procedure will work when one has a single horizon, including an asymptotically anti-de Sitter spacetime [74,75]. However, this procedure will not work for a spacetime with two horizons having differing values of |κ|. The conclusion seems to be that classically, the sign of the temperature can only be determined by appealing to the first law, and this provides us with a Gibbsian temperature. Quantum mechanically, which seems to be the only physically reliable argument provided one is prepared to contemplate non-equilibrium situations, the temperature should be taken to be positive. In other words, the temperature is not unquely defined by the metric, a conclusion also reached in [25].
The original suggestion that inner horizons should be assigned a negative temperature [1] was based not quantum field theoretic considerations, but rather on a consideration of quantum mechanical systems, such as spin systems, exhibiting population inversion [58].
Thus one might regard the total energy of a black hole as receiving contributions both from the outer and inner horizons. The inner system would then be thought of as the analogue of a spin system. This viewpoint was supported by the existence for the Kerr-Newman black hole of the modified Smarr formula (3.34), and its variation, which may be written as dM = 1 2 (T + dS + + Ω + dJ + Φ + dQ) + 1 2 (T − dS − + Ω + dJ + Φ + dQ) . (7.1) As we saw, these formulae generalise to the case of STU black holes with four electric charges. The addition of electric charges, which were not included in the discussion in [1], suggest that the posited spin system inverted population should be supplemented by the inclusion of charged states.
In the case of four-dimensional STU black holes, the generalisation of equation (7.1) may be rewitten in terms of the left-moving and right-moving sectors (see (3.69)) as with each sector contributing equally to dM . In contrast to the proposal in [1], which attempted to give a microscopic interpretation to the negative temperature on the inner horizon, here the left-moving and right-moving sectors both have positive temperatures, consistent with the proposed microscopic interpretation in terms of D-brane states [11,62].
An analogous interpretation for five-dimensional STU black holes has also been given [16].
This paper has been concerned exclusively with time-independent solutions; we have not discussed what happens to inner horizons when perturbations are considered. There is a widespread belief that in classical general relativity, generic perturbations will render Cauchy horizons, of the sort one finds inside black holes, singular. This is referred to as the Cosmic Censorship Hypothesis. There are various forms of this hypothesis, and the literature is at present rather inconclusive. A recent discussion can be found in [97]. Our motivation is largely quantum mechanical, and the relevance of these classical results to a full quantum gravitational treatment is unclear. The metric will be regular as long as A, f and R 2 are real, bounded, and twice differentiable, and in addition f and R are non-zero. We may take f , without loss of generality, to be positive. In particular, the metric is well-behaved regardless of whether A is positive, zero or negative. Asymptotic flatness requires that A and f tend to 1 as R 2 tends to infinity. In the cases we shall consider, R tends to r at infinity. We shall assume that A is positive in the interval r + < r ≤ ∞, and negative in the the interval r − < r < r + , and that it vanishes on the outer horizon r = r + and the inner horizon r = r − . We shall also assume that A has a smooth positive extension for values of r < r − . The Killing vector K = ∂/∂v is thus timelike for r + < r < ∞, lightlike at r = r + , spacelike for r − < r < r + , lightlike at r = r − and timelike for r < r − . It becomes lightlike as v tends to ±∞, and also as r tends to infinity.
If r + < r < ∞, then as v tends to +∞ we obtain future null infinity, I + . For v instead tending to −∞, we obtain past null infinity I − . As v tends to −∞ and r tends to r + we obtain the past null horizon. The Killing vector K is future-directed inside and on the boundary of this region. The inner region is bounded by a past Cauchy horizon at v = −∞ and r = r + , and a future Cauchy horizon at v = +∞ and r = r − . It has a further boundary on the inner horizon at r = r − , with −∞ < v < +∞. Thus the Killing vector K is future directed both on this inner horizon and on the outer horizon.
If one looks at radial geodesics in this spacetime, there are two conserved quantities p v and k, where and a dot denotes a derivative with respect to an affine parameter λ. Thus radially-infalling geodesics obeyṙ with k > 0 and p 2 v > k for timlike geodesics that originate at large r. The constant p v is positive. The infalling particle passes through the outer and the inner horizons before reaching a turning point at a radiusr < r − at which p 2 v = kA(r). Solving forv one findsv and so dv dr Thus one finds thatv, dv/dr and v all remain finite as the particle falls in from infinity tō r. Note thatv is always positive.
In conclusion, we note that the Killing vector K = ∂/∂v is future directed and lightlike on both the future event horizon of the exterior region, r = r + with −∞ < v < +∞, and on the inner horizon, r = r − with −∞ < v < +∞.
For the four-charge STU black holes considered in this paper, the situation when they are non-rotating is qualitatively similar to that for the Reissner-Nordström solution. The metric takes the form where The outer horizon is located at r + = µ, and the inner horizon at r − = 0. There are curvature singularities at the four locations r = −µ sinh 2 δ i , and the Carter-Penrose diagram will be similar to that for Reissner-Nordström, with the curvature singularity in the diagram occurring at the least negative of the four locations.

B STU Supergravity
The Lagrangian of the bosonic sector of four-dimensional ungauged STU supergravity can be written in the relatively simple form where the index i labelling the dilatons ϕ i and axions χ i ranges over 1 ≤ i ≤ 3. The four field strengths can be written in terms of potentials as The field strengths here are not in the same duality frame as the one we have assumed in our discussions in this paper however. To convert from (B.1) and (B.2) to the frame we are using, one would need to dualise the field strengths F 1 (2) and F 2 (2) , and if then written explicitly, the resulting Lagrangian would be rather cumbersome. Alternatively, one could simply exchange the roles of the electric and magnetic charges for the field strengths F 1 (2) and F 2 (2) , and work with (B.1) without performing any dualisations. For example, the 4charge black hole solutions that we refer to in this paper as having four electric charges would, as solutions in terms of the fields in (B.1), instead comprise two electric and two magnetic charges. (As for example, in the presentation of these solution in [64].)